metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.27D6, D12.14D10, C60.41C23, Dic30.15C22, Q8⋊D5⋊7S3, C15⋊D8⋊8C2, C15⋊21(C4○D8), Q8⋊3S3⋊2D5, C15⋊7Q16⋊7C2, C5⋊2C8.19D6, (C5×Q8).37D6, (C3×Q8).5D10, Q8.10(S3×D5), D20⋊5S3⋊3C2, (C4×S3).25D10, (S3×C10).14D4, C10.151(S3×D4), D12.D5⋊7C2, C30.203(C2×D4), D6.3(C5⋊D4), C5⋊7(Q8.7D6), C3⋊4(D4.8D10), C20.41(C22×S3), (C5×Dic3).40D4, C12.41(C22×D5), (S3×C20).13C22, C15⋊3C8.15C22, (C3×D20).15C22, (C5×D12).15C22, (Q8×C15).11C22, Dic3.22(C5⋊D4), C4.41(C2×S3×D5), (S3×C5⋊2C8)⋊6C2, (C3×Q8⋊D5)⋊5C2, C2.32(S3×C5⋊D4), C6.54(C2×C5⋊D4), (C5×Q8⋊3S3)⋊2C2, (C3×C5⋊2C8).11C22, SmallGroup(480,593)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.27D6
G = < a,b,c,d | a20=b2=d2=1, c6=a10, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a15b, dcd=c5 >
Subgroups: 604 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C4○D8, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, Dic15, C60, C60, C6×D5, S3×C10, S3×C10, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.7D6, C3×C5⋊2C8, C15⋊3C8, D5×Dic3, C15⋊D4, C3×D20, S3×C20, S3×C20, C5×D12, C5×D12, Dic30, Q8×C15, D4.8D10, S3×C5⋊2C8, C15⋊D8, D12.D5, C3×Q8⋊D5, C15⋊7Q16, D20⋊5S3, C5×Q8⋊3S3, D20.27D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q8.7D6, C2×S3×D5, D4.8D10, S3×C5⋊D4, D20.27D6
►On 240 points
Generators in S240
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 224)(2 223)(3 222)(4 221)(5 240)(6 239)(7 238)(8 237)(9 236)(10 235)(11 234)(12 233)(13 232)(14 231)(15 230)(16 229)(17 228)(18 227)(19 226)(20 225)(21 82)(22 81)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 86)(38 85)(39 84)(40 83)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(101 191)(102 190)(103 189)(104 188)(105 187)(106 186)(107 185)(108 184)(109 183)(110 182)(111 181)(112 200)(113 199)(114 198)(115 197)(116 196)(117 195)(118 194)(119 193)(120 192)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 160)(129 159)(130 158)(131 157)(132 156)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(161 209)(162 208)(163 207)(164 206)(165 205)(166 204)(167 203)(168 202)(169 201)(170 220)(171 219)(172 218)(173 217)(174 216)(175 215)(176 214)(177 213)(178 212)(179 211)(180 210)
(1 182 90 140 202 66 11 192 100 130 212 76)(2 193 91 131 203 77 12 183 81 121 213 67)(3 184 92 122 204 68 13 194 82 132 214 78)(4 195 93 133 205 79 14 185 83 123 215 69)(5 186 94 124 206 70 15 196 84 134 216 80)(6 197 95 135 207 61 16 187 85 125 217 71)(7 188 96 126 208 72 17 198 86 136 218 62)(8 199 97 137 209 63 18 189 87 127 219 73)(9 190 98 128 210 74 19 200 88 138 220 64)(10 181 99 139 211 65 20 191 89 129 201 75)(21 141 176 60 222 113 31 151 166 50 232 103)(22 152 177 51 223 104 32 142 167 41 233 114)(23 143 178 42 224 115 33 153 168 52 234 105)(24 154 179 53 225 106 34 144 169 43 235 116)(25 145 180 44 226 117 35 155 170 54 236 107)(26 156 161 55 227 108 36 146 171 45 237 118)(27 147 162 46 228 119 37 157 172 56 238 109)(28 158 163 57 229 110 38 148 173 47 239 120)(29 149 164 48 230 101 39 159 174 58 240 111)(30 160 165 59 231 112 40 150 175 49 221 102)
(1 71)(2 62)(3 73)(4 64)(5 75)(6 66)(7 77)(8 68)(9 79)(10 70)(11 61)(12 72)(13 63)(14 74)(15 65)(16 76)(17 67)(18 78)(19 69)(20 80)(21 156)(22 147)(23 158)(24 149)(25 160)(26 151)(27 142)(28 153)(29 144)(30 155)(31 146)(32 157)(33 148)(34 159)(35 150)(36 141)(37 152)(38 143)(39 154)(40 145)(41 238)(42 229)(43 240)(44 231)(45 222)(46 233)(47 224)(48 235)(49 226)(50 237)(51 228)(52 239)(53 230)(54 221)(55 232)(56 223)(57 234)(58 225)(59 236)(60 227)(81 126)(82 137)(83 128)(84 139)(85 130)(86 121)(87 132)(88 123)(89 134)(90 125)(91 136)(92 127)(93 138)(94 129)(95 140)(96 131)(97 122)(98 133)(99 124)(100 135)(101 169)(102 180)(103 171)(104 162)(105 173)(106 164)(107 175)(108 166)(109 177)(110 168)(111 179)(112 170)(113 161)(114 172)(115 163)(116 174)(117 165)(118 176)(119 167)(120 178)(181 216)(182 207)(183 218)(184 209)(185 220)(186 211)(187 202)(188 213)(189 204)(190 215)(191 206)(192 217)(193 208)(194 219)(195 210)(196 201)(197 212)(198 203)(199 214)(200 205)
G:=sub<Sym(240)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,224)(2,223)(3,222)(4,221)(5,240)(6,239)(7,238)(8,237)(9,236)(10,235)(11,234)(12,233)(13,232)(14,231)(15,230)(16,229)(17,228)(18,227)(19,226)(20,225)(21,82)(22,81)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(101,191)(102,190)(103,189)(104,188)(105,187)(106,186)(107,185)(108,184)(109,183)(110,182)(111,181)(112,200)(113,199)(114,198)(115,197)(116,196)(117,195)(118,194)(119,193)(120,192)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,220)(171,219)(172,218)(173,217)(174,216)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210), (1,182,90,140,202,66,11,192,100,130,212,76)(2,193,91,131,203,77,12,183,81,121,213,67)(3,184,92,122,204,68,13,194,82,132,214,78)(4,195,93,133,205,79,14,185,83,123,215,69)(5,186,94,124,206,70,15,196,84,134,216,80)(6,197,95,135,207,61,16,187,85,125,217,71)(7,188,96,126,208,72,17,198,86,136,218,62)(8,199,97,137,209,63,18,189,87,127,219,73)(9,190,98,128,210,74,19,200,88,138,220,64)(10,181,99,139,211,65,20,191,89,129,201,75)(21,141,176,60,222,113,31,151,166,50,232,103)(22,152,177,51,223,104,32,142,167,41,233,114)(23,143,178,42,224,115,33,153,168,52,234,105)(24,154,179,53,225,106,34,144,169,43,235,116)(25,145,180,44,226,117,35,155,170,54,236,107)(26,156,161,55,227,108,36,146,171,45,237,118)(27,147,162,46,228,119,37,157,172,56,238,109)(28,158,163,57,229,110,38,148,173,47,239,120)(29,149,164,48,230,101,39,159,174,58,240,111)(30,160,165,59,231,112,40,150,175,49,221,102), (1,71)(2,62)(3,73)(4,64)(5,75)(6,66)(7,77)(8,68)(9,79)(10,70)(11,61)(12,72)(13,63)(14,74)(15,65)(16,76)(17,67)(18,78)(19,69)(20,80)(21,156)(22,147)(23,158)(24,149)(25,160)(26,151)(27,142)(28,153)(29,144)(30,155)(31,146)(32,157)(33,148)(34,159)(35,150)(36,141)(37,152)(38,143)(39,154)(40,145)(41,238)(42,229)(43,240)(44,231)(45,222)(46,233)(47,224)(48,235)(49,226)(50,237)(51,228)(52,239)(53,230)(54,221)(55,232)(56,223)(57,234)(58,225)(59,236)(60,227)(81,126)(82,137)(83,128)(84,139)(85,130)(86,121)(87,132)(88,123)(89,134)(90,125)(91,136)(92,127)(93,138)(94,129)(95,140)(96,131)(97,122)(98,133)(99,124)(100,135)(101,169)(102,180)(103,171)(104,162)(105,173)(106,164)(107,175)(108,166)(109,177)(110,168)(111,179)(112,170)(113,161)(114,172)(115,163)(116,174)(117,165)(118,176)(119,167)(120,178)(181,216)(182,207)(183,218)(184,209)(185,220)(186,211)(187,202)(188,213)(189,204)(190,215)(191,206)(192,217)(193,208)(194,219)(195,210)(196,201)(197,212)(198,203)(199,214)(200,205)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,224)(2,223)(3,222)(4,221)(5,240)(6,239)(7,238)(8,237)(9,236)(10,235)(11,234)(12,233)(13,232)(14,231)(15,230)(16,229)(17,228)(18,227)(19,226)(20,225)(21,82)(22,81)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(101,191)(102,190)(103,189)(104,188)(105,187)(106,186)(107,185)(108,184)(109,183)(110,182)(111,181)(112,200)(113,199)(114,198)(115,197)(116,196)(117,195)(118,194)(119,193)(120,192)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,220)(171,219)(172,218)(173,217)(174,216)(175,215)(176,214)(177,213)(178,212)(179,211)(180,210), (1,182,90,140,202,66,11,192,100,130,212,76)(2,193,91,131,203,77,12,183,81,121,213,67)(3,184,92,122,204,68,13,194,82,132,214,78)(4,195,93,133,205,79,14,185,83,123,215,69)(5,186,94,124,206,70,15,196,84,134,216,80)(6,197,95,135,207,61,16,187,85,125,217,71)(7,188,96,126,208,72,17,198,86,136,218,62)(8,199,97,137,209,63,18,189,87,127,219,73)(9,190,98,128,210,74,19,200,88,138,220,64)(10,181,99,139,211,65,20,191,89,129,201,75)(21,141,176,60,222,113,31,151,166,50,232,103)(22,152,177,51,223,104,32,142,167,41,233,114)(23,143,178,42,224,115,33,153,168,52,234,105)(24,154,179,53,225,106,34,144,169,43,235,116)(25,145,180,44,226,117,35,155,170,54,236,107)(26,156,161,55,227,108,36,146,171,45,237,118)(27,147,162,46,228,119,37,157,172,56,238,109)(28,158,163,57,229,110,38,148,173,47,239,120)(29,149,164,48,230,101,39,159,174,58,240,111)(30,160,165,59,231,112,40,150,175,49,221,102), (1,71)(2,62)(3,73)(4,64)(5,75)(6,66)(7,77)(8,68)(9,79)(10,70)(11,61)(12,72)(13,63)(14,74)(15,65)(16,76)(17,67)(18,78)(19,69)(20,80)(21,156)(22,147)(23,158)(24,149)(25,160)(26,151)(27,142)(28,153)(29,144)(30,155)(31,146)(32,157)(33,148)(34,159)(35,150)(36,141)(37,152)(38,143)(39,154)(40,145)(41,238)(42,229)(43,240)(44,231)(45,222)(46,233)(47,224)(48,235)(49,226)(50,237)(51,228)(52,239)(53,230)(54,221)(55,232)(56,223)(57,234)(58,225)(59,236)(60,227)(81,126)(82,137)(83,128)(84,139)(85,130)(86,121)(87,132)(88,123)(89,134)(90,125)(91,136)(92,127)(93,138)(94,129)(95,140)(96,131)(97,122)(98,133)(99,124)(100,135)(101,169)(102,180)(103,171)(104,162)(105,173)(106,164)(107,175)(108,166)(109,177)(110,168)(111,179)(112,170)(113,161)(114,172)(115,163)(116,174)(117,165)(118,176)(119,167)(120,178)(181,216)(182,207)(183,218)(184,209)(185,220)(186,211)(187,202)(188,213)(189,204)(190,215)(191,206)(192,217)(193,208)(194,219)(195,210)(196,201)(197,212)(198,203)(199,214)(200,205) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,224),(2,223),(3,222),(4,221),(5,240),(6,239),(7,238),(8,237),(9,236),(10,235),(11,234),(12,233),(13,232),(14,231),(15,230),(16,229),(17,228),(18,227),(19,226),(20,225),(21,82),(22,81),(23,100),(24,99),(25,98),(26,97),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(37,86),(38,85),(39,84),(40,83),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(101,191),(102,190),(103,189),(104,188),(105,187),(106,186),(107,185),(108,184),(109,183),(110,182),(111,181),(112,200),(113,199),(114,198),(115,197),(116,196),(117,195),(118,194),(119,193),(120,192),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,160),(129,159),(130,158),(131,157),(132,156),(133,155),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148),(161,209),(162,208),(163,207),(164,206),(165,205),(166,204),(167,203),(168,202),(169,201),(170,220),(171,219),(172,218),(173,217),(174,216),(175,215),(176,214),(177,213),(178,212),(179,211),(180,210)], [(1,182,90,140,202,66,11,192,100,130,212,76),(2,193,91,131,203,77,12,183,81,121,213,67),(3,184,92,122,204,68,13,194,82,132,214,78),(4,195,93,133,205,79,14,185,83,123,215,69),(5,186,94,124,206,70,15,196,84,134,216,80),(6,197,95,135,207,61,16,187,85,125,217,71),(7,188,96,126,208,72,17,198,86,136,218,62),(8,199,97,137,209,63,18,189,87,127,219,73),(9,190,98,128,210,74,19,200,88,138,220,64),(10,181,99,139,211,65,20,191,89,129,201,75),(21,141,176,60,222,113,31,151,166,50,232,103),(22,152,177,51,223,104,32,142,167,41,233,114),(23,143,178,42,224,115,33,153,168,52,234,105),(24,154,179,53,225,106,34,144,169,43,235,116),(25,145,180,44,226,117,35,155,170,54,236,107),(26,156,161,55,227,108,36,146,171,45,237,118),(27,147,162,46,228,119,37,157,172,56,238,109),(28,158,163,57,229,110,38,148,173,47,239,120),(29,149,164,48,230,101,39,159,174,58,240,111),(30,160,165,59,231,112,40,150,175,49,221,102)], [(1,71),(2,62),(3,73),(4,64),(5,75),(6,66),(7,77),(8,68),(9,79),(10,70),(11,61),(12,72),(13,63),(14,74),(15,65),(16,76),(17,67),(18,78),(19,69),(20,80),(21,156),(22,147),(23,158),(24,149),(25,160),(26,151),(27,142),(28,153),(29,144),(30,155),(31,146),(32,157),(33,148),(34,159),(35,150),(36,141),(37,152),(38,143),(39,154),(40,145),(41,238),(42,229),(43,240),(44,231),(45,222),(46,233),(47,224),(48,235),(49,226),(50,237),(51,228),(52,239),(53,230),(54,221),(55,232),(56,223),(57,234),(58,225),(59,236),(60,227),(81,126),(82,137),(83,128),(84,139),(85,130),(86,121),(87,132),(88,123),(89,134),(90,125),(91,136),(92,127),(93,138),(94,129),(95,140),(96,131),(97,122),(98,133),(99,124),(100,135),(101,169),(102,180),(103,171),(104,162),(105,173),(106,164),(107,175),(108,166),(109,177),(110,168),(111,179),(112,170),(113,161),(114,172),(115,163),(116,174),(117,165),(118,176),(119,167),(120,178),(181,216),(182,207),(183,218),(184,209),(185,220),(186,211),(187,202),(188,213),(189,204),(190,215),(191,206),(192,217),(193,208),(194,219),(195,210),(196,201),(197,212),(198,203),(199,214),(200,205)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 12 | 20 | 2 | 2 | 3 | 3 | 4 | 60 | 2 | 2 | 2 | 40 | 10 | 10 | 30 | 30 | 2 | 2 | 12 | ··· | 12 | 4 | 8 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | Q8.7D6 | C2×S3×D5 | D4.8D10 | S3×C5⋊D4 | D20.27D6 |
| kernel | D20.27D6 | S3×C5⋊2C8 | C15⋊D8 | D12.D5 | C3×Q8⋊D5 | C15⋊7Q16 | D20⋊5S3 | C5×Q8⋊3S3 | Q8⋊D5 | C5×Dic3 | S3×C10 | Q8⋊3S3 | C5⋊2C8 | D20 | C5×Q8 | C4×S3 | D12 | C3×Q8 | C15 | Dic3 | D6 | C10 | Q8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.27D6 ►in GL6(𝔽241)
| 177 | 0 | 0 | 0 | 0 | 0 |
| 177 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 181 | 0 | 0 | 0 | 0 |
| 19 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 71 | 0 | 0 |
| 0 | 0 | 36 | 133 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 239 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 64 | 113 | 0 | 0 | 0 | 0 |
| 64 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [177,177,0,0,0,0,0,64,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,19,0,0,0,0,181,211,0,0,0,0,0,0,108,36,0,0,0,0,71,133,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,1,0,0,0,0,239,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[64,64,0,0,0,0,113,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,1,1] >;
D20.27D6 in GAP, Magma, Sage, TeX
D_{20}._{27}D_6 % in TeX
G:=Group("D20.27D6"); // GroupNames label
G:=SmallGroup(480,593);
// by ID
G=gap.SmallGroup(480,593);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^6=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^15*b,d*c*d=c^5>;
// generators/relations